Vibration is a physical phenomena characterized by oscillatory deformation of an elastic body about a position of equilibrium. The basic physical concepts involved in vibratory motion are fairly simple. Deformation of an elastic body in a first direction by the application of an external force provides the elastic body with an initial amount of mechanical energy in the form of potential energy (p.sub.1). Removal of the external force results in movement of the deformed elastic body from the high-energy deformed position towards the low-energy equilibrium position. Movement of the elastic body from the deformed position to the equilibrium position intrinsically results in the irreversible dissipation of a portion of the mechanical energy (d1) and conversion of the remaining mechanical energy (p.sub.1 -d.sub.1) from potential energy to kinetic energy (k.sub.1). The thus converted kinetic energy causes the elastic body to move past the equilibrium position and result in deformation of the elastic body in a second direction. Deformation of the elastic body in the second direction intrinsically results in the irreversible dissipation of a second portion of the mechanical energy (d.sub.2) and reconversion of the remaining mechanical energy (p.sub.1 -(d.sub.1 +d.sub.2)) from kinetic energy back into potential energy. Movement of the elastic body reverses when the kinetic energy is completely converted to potential energy (p.sub.2). The thus deformed elastic body possess an amount of potential energy (p.sub.2) equal to the initial amount of potential energy (p.sub.1) minus the amount of energy irreversibly dissipated (d.sub.1 +d.sub.2). Oscillation of the elastic body about the equilibrium position continues until the cumulative amounts of energy irreversibly dissipated (.SIGMA.d) equals the amount of mechanical energy originally provided to the elastic body (p.sub.1).
Vibration of a deformed elastic body can be perpetuated by periodically adding sufficient mechanical energy to the vibrating body to compensate for the energy lost through intrinsic dissipation.
The irreversible dissipation of mechanical energy from a vibrating elastic body is an intrinsic phenomena commonly referred to as damping. Damping is believed to result from a variety of energy loss mechanisms such as (i) the conversion of mechanical energy to heat through internal friction within the elastic body [hysteresis], (ii) the conversion of mechanical energy to heat through friction caused by the rubbing of one component of the elastic body against another, (iii) the transfer of mechanical energy from the vibrating elastic body to adjacent structural components, (iv) the transfer of mechanical energy from the vibrating elastic body to the environment through acoustic radiation, (v) the conversion of mechanical energy to heat through a viscous response either inherent in the system or subsequently added to the system.
The energy dissipation mechanisms themselves are very complex and dependant upon a great number of factors including specifically, but not exclusively: the composition of the elastic body, the crystallinity of the elastic body, the geometry of the elastic body, the temperature of the elastic body, the initial strain placed upon the elastic body, the amount of any preload placed upon the elastic body, the interrelationship between the elastic body and other bodies, the amplitude and frequency of the vibration, and the amount of viscous response.
Due to the variety of dissipative mechanisms and the internal complexity of those mechanism, it is extremely difficult to accurately predict the damping effect of a given material. However, despite such complexities, the material damping behavior for harmonic motion caused by normal stress/strain can generally be represented by the complex equation set forth below as Equation (1): EQU .sigma.=E(1+i .PHI.).andgate. (1)
wherein: .sigma. is normal stress (force/area)
E is normal modulus (dimensionless) PA1 .andgate. is normal loss factor (dimensionless) PA1 .PHI.=iwe.sup.iwt where: w is circular frequency t is time. PA1 G is shear modulus (dimensionless) PA1 .andgate.' is shear loss factor and is approximately equal to .andgate. PA1 .GAMMA. is shear strain.
It is noted that similar considerations apply to the material damping behavior for harmonic motion caused by shear stress/strain and can be represented by the complex equation set forth below as Equation (2). EQU .delta.=G(1+i.andgate.').GAMMA. (2)
wherein: .delta. is shear stress
Based upon these theoretical equations, the damping behavior of a material is dependent upon the modulus and loss factor of the material. Hence, knowledge of the modulus and/or loss factor of a material permits an assessment of the damping behavior of a material.
The modulus and loss factor variables of a damping material are highly dependent upon the temperature of the damping material and the vibration frequency. Hence, when representing experimental data of the modulus and/or loss factor of a material the representation must take into consideration the temperature and frequency at which such data was obtained.
Experimental data of the modulus and/or loss factor of a material is typically represented in the form of a reduced-temperature nomograph such as that depicted in FIG. 1. Reduced-temperature nomographs for a variety of damping materials are readily available from a number of sources including Soovere, J. and Drake, M. L., Aerospace Structures Technology Damping Design Guide, AFWAL-TR-843089, Volumes 1-3, December 1985 and Ferry, J. D., Viscoelastic Properties of Polymers, John Wiley & Sons, New York 1961. Depiction of the modulus and/or loss factor values of a material in the form of a reduced-temperature nomograph greatly simplifies determination of the modulus and loss factor values of a material by providing for display of the modulus and loss factor values verses both temperature and frequency on a single graph.
Use of a reduced-temperature nomograph to determine the modulus and loss factor of a material includes the steps of: (Step 1) locating the vibration frequency of concern on the right hand vertical axis of the nomograph, (Step 2) moving horizontally along that frequency line to the temperature isotherm representing the temperature of concern, (Step 3) moving vertically from that temperature/frequency coordinate to the modulus curve, (Step 4) moving horizontally from the modulus curve coordinate to the left vertical axis, (Step 5) reading the value of the modulus from the modulus scale provided on the left vertical axis, (Step 6) relocating the temperature/frequency coordinate found in step 2, (Step 7) moving vertically from the temperature/frequency coordinate to the loss factor curve, (Step 8) moving horizontally from the loss factor curve coordinate to the left vertical axis, and (Step 9) reading the value of the loss factor from the loss factor scale provided on the left vertical axis.
Advent of the reduced temperature nomograph constitutes a tremendous advancement over the previously employed method of determining modulus and loss factor based upon separate temperature and frequency graphs. However, even with the increased simplicity offered by reduced-temperature nomographs, many individuals, particularly those with a limited scientific background, still find it difficult to determine the modulus and loss factor of a material.
Accordingly, a substantial need exists for a simpler method of determining the modulus and loss factor of a damping material based upon temperature and frequency variables.